1

1 A MODEL TO SIMULATE THE CONTRIBUTION OF FIBRE REINFORCEMENT FOR THE 2

PUNCHING RESISTANCE OF RC SLABS 3

4

5

Bernardo N. Moraes Neto 1, Joaquim A.O. Barros 2, Guilherme S.S.A. Melo 3 6

7

1 University of Minho/University of Brasília-UnB, Dep. Civil Eng., School of Eng., Guimarães, Portugal, 8

bnmn@hotmail.com. 9

2 University of Minho, ISISE, Dep. Civil Eng., School of Eng., Campus de Azurém, Guimarães, Portugal, 10

barros@civil.uminho.pt. 11

3 University of Brasília-UnB, Dep. Civil Eng., Campus of Darcy Ribeiro, Brasília, Brasil, melog@unb.br. 12

13

14

Abstract 15

In this paper analytical formulations are developed for the prediction of the punching resistance of flat slabs 16

of steel fibre reinforced concrete (SFRC) flexurally reinforced with steel bars. By performing statistical 17

analysis with a database that collects experimental results on the characterization of the post-cracking 18

behaviour of SFRC, equations are determined for the evaluation of the residual flexural tensile strength 19

parameters (fRi) from fundamental data that characterize steel fibres. The fRi strength parameters proposed by 20

CEB-FIP 2010 were used for the definition of the stress-crack width law (σ-w) that simulates the fibre 21

reinforcement mechanisms in cement based materials. In the second part of the paper is described an 22

2

analytical formulation based on the concepts proposed by Muttoni and Ruiz, where the σ-w law is 23

conveniently integrated for the simulation of the contribution of steel fibres for the punching resistance of 24

SFRC slabs. By using a database composed of 154 punching tests with SFRC slabs, the good predictive 25

performance of the developed proposal is demonstrated. The good performance of this model is also 26

evidenced by comparing its predictions to those from other models. 27

28

Keywords: Reinforced concrete, Flat slab, Punching, Steel fibre reinforced concrete, Analytical models. 29

30

INTRODUCTION 31

Recent experimental programs have shown the possibility of building structural systems based on flat slabs 32

of steel fibre reinforced concrete (SFRC) supported on reinforced concrete (RC) columns (Espion 2004; 33

Mandl 2008; Destrée et al. 2009; Destrée and Mandl 2008; Barros et al. 2012). This type of slabs is 34

generally designated by Elevated Steel Fibre Reinforced Concrete Slabs (ESFRC), and it includes a 35

minimum continuity rebars, also referred as anti-progressive collapse rebars, placed in the bottom of the slab 36

in the alignment of the columns (Sasani and Sagiroglu 2008). The results obtained in these experimental tests 37

have demonstrated that this construction system fulfills the structural exigencies required for residential 38

buildings, and is a competitive alternative to the availabe conventional construction methods. However, a 39

reliable acceptance of this innovative construction system also requires the existence of design guidelines 40

that can predict its structural behaviour with high accuracy, namely the punching resistance, since punching 41

failure is quite brittle and, in general, conducts to the global collapse of a building (Gardner et al. 2002). 42

In terms of punching resistance of conventionally reinforced concrete flat slabs, in general, the actual design 43

standards, such is the case of ACI 318 (2008), BS 8110 (1985), EC2 (2004), and CEB-FIP Model Code 44

1990, adopt the approach that the ultimate punching resistance, VR,d, is obtained adding the parcel due to the 45

concrete resistance, VR,cd, to the term that simulates the contribution of shear reinforcement, VR,sd, e.g., 46

VR,d=VR,cd+VR,sd. For slabs without shear reinforcement, VR,d=VR,cd, and the design procedure for punching is 47

based on the verification that the nominal shear stress, νN,d, in two or more critical sections around the 48

column does not exceed the nominal shear strength, νR,d, e.g., νR,d≥νN,d. 49

3

Recently, the CEB-FIP Model Code 2010 proposed recommendations for the evaluation of the flexural and 50

shear resistance of members made by fibre reinforced concrete (FRC). These recommendations are 51

supported on residual flexural tensile strength parameters, fRi, that characterize the post-cracking behaviour 52

of FRC, and are determined from three point bending tests with notched FRC beams. The full definition and 53

the strategy to obtain fRi from experimental tests, as well as the aforementioned recommendations, are 54

described in the following sections.The CEB-FIP Model Code 2010 has also proposed a very simple 55

approach to simulate the contribution of fibre reinforcement for the punching resistance of FRC flat slabs, 56

VR,fd. Several studies have been done with the purpose of developing a design approach for the prediction of 57

the contribution of fibre reinforcement for the punching resistance of SFRC slabs (Narayanan and Darwish 58

1987; Harajli et al. 1995; Muttoni and Ruiz 2010; Michels et al. 2012), but the predictive performance of 59

these models is generally limited to a relatively small number of punching tests, and the contribution of fibre 60

reinforcement is not based on the most recent knowledge about modelling the post-cracking behaviour of this 61

composite. 62

63

In the present paper a design formulation is proposed for the evaluation of the punching resistance of SFRC 64

slabs. This model is based on the principles proposed by Muttoni and Ruiz (2010), where a stress-crack 65

width relationship is used to simulate the contribution of the fibre reinforcement mechanisms for the 66

punching resistance of SFRC flat slabs, VR,f. The VR,f is determined from the fRi parameters proposed by CEB-67

FIP Model Code 2010. 68

69

DESIGN FORMULATIONS PROPOSED BY STANDARD CODES 70

Design formulation for flat slabs only reinforced with conventional steel bars 71

ACI 318 (2008) 72

According to ACI 318, the punching resistance of the type of slabs analysed in the present paper is obtained 73

from: 74

d,Sd,R VV ≥⋅φ (1) 75

where, 76

4

dbV d,Rd,R ⋅⋅= 0ν (2) 77

with 78

+⋅⋅⋅

+⋅=

31

6

6

21

6 0

csc

c

cd,R

f;

b

df;

fmin

αβ

ν [MPa, mm] (3)

79

In Eq. (2) b0 is the perimeter corresponding to the formation of the punching failure surface, assumed 80

localized at a distance α=0.5 from the external face of the column, and with the geometric configuration 81

represented in Figure 1a.In Eq. (3) βc is the ratio between the larger and the smaller edge of the column’s 82

cross section, αs=3.32 for columns located in the interior of the building (assumed centrically loaded), such 83

is the case treated in the present work, fc is the average concrete compressive strength evaluated using 84

cylinder specimens, and d is the internal arm of the longitudinal tensile reinforcement of the slab’s cross 85

section. 86

87

CEB-FIP MODEL CODE 1990 88

The CEB-FIP Model Code 1990 recommends that the punching resistance of a RC slab without shear 89

reinforcement should be determined from the following equation: 90

duV d,Rd,R ⋅⋅= 1ν [MPa, mm] (4) 91

with 92

( ) 31

100120 cd,R f. ⋅⋅⋅⋅= ρξν [MPa, mm] (5)

93

where 94

d

2001+=ξ [mm]; (6)

yx ρρρ ⋅=

(7)

95

The ξ parameter in Eq. (6) aims to simulate the size effect. The reinforcement ratio of the tensile flexural 96

reinforcement, ρ, is calculated from Eq. (7) by considering the reinforcement ratio ρx and ρy in the two main 97

orthogonal directions (x and y). For the evaluation of ρx and ρy, a width of the slab cross section equal to 98

5

e+6·d (for the columns of square cross section), or equal to 2·rc+6·d (for the columns of circular cross 99

section), is considered, Figure 1. 100

101

EC2 (2004) 102

In the Eurocode 2, EC2, it is assumed that the punching resistance of RC slabs without shear reinforcement 103

can be estimated by the following equation: 104

duV d,Rd,R ⋅⋅= 1ν (8) 105

with 106

( )

⋅⋅⋅⋅⋅⋅= 2

123

31

0350100 ccc,Rdd,R fk.;fkCmax ρν [MPa, mm] (9)

107

where k is defined as ξ (Eq. (6)), but a maximum limit of 2.0 is imposed to k, while ρ is obtained from Eq. 108

(7) with an upper limit of 0.02. In Eq. (8) the critical perimeter, u1, is localized at 2d from the external 109

surface of the column (α=2), and can assume the configurations represented in Figures 1b and 1c. In Eq. (9) 110

CRd,c=0.18/γc. 111

112

CEB-FIP Model Code 2010 113

According to the CEB-FIP Model Code 2010, the punching resistance of RC slabs without shear 114

reinforcement, VR,d=VR,cd, is determined from the following equation: 115

dbf

kVVc

ccd,Rd,R ⋅⋅⋅== 0γψ [MPa, mm] (10)

116

where b0 is the critical punching perimeter at a distance α=0.5 from the external surface of the column, as 117

represented in Figures 1b and 1c. The kψ parameter depends of the rotation of the slab, and is determined 118

from the following equation: 119

609051

1.

kd..k

dg≤

⋅⋅⋅+=

ψψ [mm] (11)

120

where kdg parameter simulates the aggregate interlock: 121

6

75016

32.

dk

gdg ≥

+= [mm] (12)

122

being dg the maximum diameter of the aggregates. 123

The rotation of the slab, ψ, (Figure 2b) required to determine the kψ parameter, is evaluated according to the 124

approach II indicated in CEB-FIP Model Code 2010, by applying the following equation: 125

51

51.

Rd

sd

s

yds

m

m

E

f

d

r.

⋅⋅⋅=ψ (13)

126

where rs indicates the position, in relation to the axis of the column, at which the radial bending moment, mr, 127

is null (Figure 2a). The value of rs can be considered equal to 0.22·Ls (Ls is replaced by Ls,x for the analysis in 128

x direction, and Ls is replaced by Ls,y for the analysis in y direction, Figure 2c) in slabs where the Ls,x/Ls,y ratio 129

pertains to the interval [0.5 – 2.0]. In Eq. (13) the msd=VS,d/8 (Johansen 1962) and mRd represents the design 130

value of the actuating and resisting bending moment, respectively. Both msd and mRd are evaluated for a slab 131

strip of a width of bs=1.5·(rs,x·rs,y)0.5≤Ls,min, where rs,x and rs,y is the rs in x and y direction, respectively, and 132

Ls,min is the minimum value between Ls,x and Ls,y, Figue 2c. The strategy to evaluate mRd will be discussed in a 133

posterior section. 134

135

Design formulation for FRC flat slabs flexurally reinforced with conventional steel bars - CEB-FIP 136

Model Code 2010 137

For SFRC slabs, the CEB-FIP Model Code 2010 recommends the following equation for the evaluation of 138

the punching resistance: 139

fd,Rcd,Rd,R VVV += (14)

140

where VR,cd is calculated according to Eq. (10), and the parcel corresponding to the contribution of fibre 141

reinforcement, VR,fd, is evaluated from: 142

dbf

VF

Ftukfd,R ⋅⋅= 0γ

(15)

143

being, 144

7

( ) 02.05.05.2 13 ≥⋅+⋅−⋅−= RRFtsu

FtsFtu fffw

ff ; 1450 RFts f.f ⋅= (16)

145

In Eq. (15) fFtuk represents the characteristic value of the ultimate residual flexural tensile strength, which is 146

calculated for an ultimate crack width (wu) of 1.5 mm. When the SFRC slab also includes conventional 147

flexural reinforcement, the CEB-FIP Model Code 2010 suggests the use of wu=ψ d/6, where ψ is calculated 148

from Eq. (13). The fFts in Eq. (16) represents the residual flexural tensile strength for the verifications of 149

serviceability limit states. The fRi (i=1 and 3) parameters indicated in Eq. (16) represent the residual flexural 150

tensile strength parameters of FRC, and are determined from the load versus Crack Mouth Opening 151

Displacement (CMOD) registered in three point notched beam bending tests, by applying the following 152

equation: 153

22

3

sp

RiRi

hb

LFf

⋅⋅⋅⋅= (17)

154

where FRi is the force corresponding to CMODi, and L (500 mm), b (150 mm) and hsp (125 mm) are the free 155

span, the width of the cross section and the distance from the tip of the notch to the top surface of the beam, 156

respectively (CEB-FIP Model Code 2010). 157

158

ASSESSMENT OF THE PREDICTIVE PERFORMANCE OF DESIGN FORMULATIONS 159

PROPOSED BY STANDARD CODES 160

Introduction 161

The predictive performance of the the formulations proposed by standard codes, described in a previous 162

Section, is evaluated in terms of the λ=Vexp/Vthe parameter, where Vexp is the punching failure load registered 163

experimentally, and Vthe=VR=VR,c is the punching failure load estimated according to the analytical 164

formulations of the considered standard codes. The purpose of this section is to determine the formulation 165

that best predicts the punching failure load of RC flat slabs (VR=VR,c), in order to be adopted with the model 166

proposed in the present work to estimate the failure load of SFRC flat slabs that are also flexurally reinforced 167

with steel bars, e.g. Vthe=VR=VR,c+VR,f. This assessment was executed by considering a database of punching 168

tests with flat slabs described in the following section. 169

170

8

Database (DB) 171

A database (DB) composed by 154 slabs submitted to punching test configuration was built, 137 of them 172

were reinforced with longitudinal steel bars/grids in order to avoid the occurrence of flexural failure mode. 173

None of these slabs has conventional shear/punching reinforcement. However, 105 slabs composing the DB 174

were made by SFRC. In terms of concrete compressive strength, fcm, the DB is composed of slabs with fcm in 175

the range of 14 to 93 MPa, so a quite high interval exists for a parameter that has a relevant impact on the 176

punching resistance of concrete slabs. For the slabs that were flexurally reinforced with steel bars, the 177

internal arm of this reinforcement (d, Figure 2) has varied from 13 mm to 180 mm, while the reinforcement 178

ratio (ρ) is in the interval 0.4 to 2.75%. In the SFRC slabs, “hooked”, “ twisted”, “ crimped”, “ corrugated”, 179

“paddle” and other types of fibres were used, with an aspect-ratio that varied from 20 to 100, and in a 180

volume percentage ≤2%. In some of the SFRC slabs (6 specimens), the SFRC was only applied in a region 181

around the loaded area (that represents the position of the column), considered the region where punching 182

failure could occur. In terms of loading conditions, all the slabs of the DB were submitted to a load 183

distributed in a certain area of the slab without transferring any bending moments from the loading device to 184

the slab. In the tests of the DB, the columns were simulated by a RC element monolithically connected to the 185

slab or applying steel plates, or even introducing a semi-spherical device in between the piston of the 186

actuator and the tested slab. The cross section of the columns and steel plates was square or circular. To 187

avoid results that can compromise the reliability of this statistical analysis, the slabs with a thickness lower 188

than 80 mm were discarded, since an eventual influence of size effect can have a detrimental consequence on 189

this study. Furthermore, the slabs where the concrete compressive strength has decreased more than 15% in 190

consequence of the addition of fibres were also neglected, since this decrease reveals that the SFRC mix 191

composition was not properly designed. Further details about the DB can be found elsewhere (Moraes Neto 192

2013). 193

In this section, slabs only reinforced with steel bars are considered for the assessment of the predictive 194

performance of the formulations proposed by standard codes, described in the previous chapter. 195

196

General statistical analysis procedures 197

9

The performance of the formulations proposed by the considered standard codes for the prediction of the 198

punching resistance of RC slabs is appraised using the collected data registered in the DB. For each proposal, 199

the obtained values of Vthe are compared with Vexp and a λ factor corresponding to the Vexp/Vthe ratio is 200

evaluated. The values of λ were classified according to the modified version of the of the Demerit Points 201

Classification (DPC) proposed by Collins (2001), where a penalty (PEN) is assigned to each range of λ 202

parameter according to Table 1, and the total of penalties (Total PEN) determines the performance of the 203

proposal. The penalty is a weighting factor determined from statistical analysis that takes into account safety, 204

accuracy and economic aspects (Collins 2001). According to this strategy, the proposal with the minimum 205

total of penalties is the best one under this framework. 206

In next section the models in analysis are designated as MODi (i=1 up to 4), with the corresponding 207

formulation assigned in the footnote of Table 2. 208

In the analysis performed, unit value was assumed for all the safety factors (such is the case of γc and γF in 209

equations (10) and (15), respectively) considered by the formulations, and average values were adopted for 210

the properties of the materials (such as: fc, fct, fR1, fR3, fFts, fFtu), since the analytical predictions will be 211

compared to the experimental results. Furthermore, in the evaluation of Eq. (10) of the model proposed by 212

CEB-FIP Model Code 2010, the Eqs. (11) and (12) that define the punching failure criterion are replaced by 213

the following ones: 214

dgkd

/k

⋅⋅+=

ψψ 1

43 [rad, mm] (18)

gdg d

k+

=16

15 [mm] (19)

215

in order to take into account that average values are now considered (Muttoni 2008). 216

217

Results 218

The results presented in the present section assess the performance of the formulations described in Section 219

“Design formulation for flat slabs only reinforced with conventional steel bars” for the prediction of the 220

punching failure load of flat slabs only reinforced with longitudinal bars, e.g. Vthe=VR=VR,c. The results are 221

presented in Table 2, and from the analysis it can be concluded that MOD4, corresponding to the one 222

proposed by CEB-FIP Model Code 2010, has best predicted the punching failure load registered 223

10

experimentally, with the lowest COV. Furthermore, it is the one with the largest number of predictions in the 224

appropriate safety interval according to the DPC (Table 1), e.g. 21 samples with λ∈[0.85-1.15[. Therefore, it 225

will be selected for the evaluation of the VR,c in the context of the formulation to be developed for the 226

prediction of the punching failure model of SFRC flat slabs flexurally reinforced with steel bars, Eq. (14). 227

228

PRACTICAL PROPOSAL FOR THE ESTIMATION OF THE RESIDUAL FLEXURAL TENSILE 229

STRENGTH PARAMETERS, fRi 230

Introduction 231

The predictive performance of the model proposed in the present work for the evaluation of the punching 232

failure load of SFRC flat slabs will be assessed by comparing the estimated results with those available in the 233

DB described in Section “Database (CB)”. As already indicated, the contribution of fibre reinforcement for 234

the punching resistance is simulated by using the concept fRi, however, these values are not available in the 235

majority of the works composing the DB. Therefore, to apply the proposed model to the tests composing the 236

DB, another database was built by collecting results (fRi) of the characterization of the post-cracking flexural 237

behaviour of SFRC according to the recommendations of CEB-FIP Model Code 2010. Since the fibre 238

volume percentage, Vf, and fibre aspect ratio, (l f/df), (quotient between fibre length, l f, and fibre diameter, df) 239

are practically the unique common information available in the works forming the DB of the punching tests, 240

the statistical analysis performed with the collected data for the characterization of the post-cracking 241

behaviour of SFRC was governed by the criterion of deriving equations for the fRi dependent on the Vf and 242

l f/df. The authors are aware that this is a quite simple approach to simulate the fibre reinforcement 243

mechanisms, since other variables like the fibre-matrix bond strength, fibre inclination and fibre embedment 244

length influence the values of fRi (Cunha et al. 2010), but this information is not available in those works. 245

Therefore, a relatively large scatter of results is naturally expected for the relationships fRi-(Vf, l f/df), but 246

actually this is the unique possibility of considering the fibre reinforcement mechanisms according to the 247

CEB-FIP Model Code 2010 for the prediction of the punching failure load of the slabs collected in the DB by 248

applying the proposed model. Preliminary statistical analysis by considering also the bond strength was also 249

carried out (Moraes Neto 2013), but the obtained results have revealed that by also adopting these 250

parameters, the dispersion of the results increase significantly, since a large scatter of bond strength values 251

11

exists in the bibliography. Taking this into account, the statistical analysis was carried out in order to derive 252

equations that conduct to safe predictions. 253

254

Database (DB) and procedures for the analysis 255

A database composed of 69 results from three point notched SFRC beam bending tests was collected 256

(Moraes Neto 2013). The analysis of the DB has indicated that fibre reinforcement index, Vf⋅l f/df, is the most 257

influential parameter on the fRi values. Taking into account the geometric characteristics and volume 258

percentage of the steel fibres most used in the available experimental programs of punching SFRC slabs, this 259

database is restricted to the tests with concrete reinforced with hooked ends steel fibres, in a volume 260

percentage ranging from 0.13% to 1.25% and with fibre aspect-ratio in the interval 50 to 80. 261

In the first step of the analysis of the information available in the DB, relationships between fRi and Vf⋅l f/df 262

were established (Moraes Neto 2013). The predictive performance of the equations were then evaluated in 263

terms of the λi=fRi,exp/fRi,the parameter, where fRi,exp and fRi,the is, respectively, the residual flexural strength 264

parameter recorded experimentally (available in the DB) and estimated according to the obtained fRi-Vf⋅l f/df 265

relationships. The predictive performance of these relationships was also appraised by using a modified 266

version of the DPC, where a penalty is assigned to each range of λi parameter according to Table 1, and the 267

total of penalties determines the performance of the fRi-Vf⋅l f/df relationship. To assure stable predictions, the 268

statistical analysis was executed in order to provide average values for λ1=fR1,exp/fR1,the and λ2,the in the lower 269

bound of the interval considered as “conservative” (Table 1), which assures safe predictions in terms of 270

design philosophy. 271

272

Assessement of the predictive performance of the fRi -Vf⋅⋅⋅⋅lf/df relationship 273

Analyzing the results of the DB it was realized that the fRi-Vf⋅l f/df function that best fit the results is of the 274

type (Moraes Neto 2013), ( ) jcfffjRi d/lVkf ⋅⋅= . To derive the kj and the cj values (j=1 and 2), a 275

parametric analysis was executed (Moraes Neto 2013) in order to obtain the best compromise in terms of the 276

lowest R-squared values (R2) of the fitting process and the lowest total penalties according to the modified 277

DPC, having resulted the following equations: 278

279

12

801

11 57

.

f

ff

c

f

ffR d

lV.

d

lVkf

⋅⋅=

⋅⋅= [MPa] (20)

702

23 06

.

f

ff

c

f

ffR d

lV.

d

lVkf

⋅⋅=

⋅⋅= [MPa] (21)

1133 850 RRR f.fkf ⋅=⋅= (22)

280

Eq. (22) shows a tendency for a linear relationship between fR1 and fR3, which was already pointed out in a 281

previous work (Barros et al. 2005). 282

The predictive performance of Eqs. (20) and (21) was assessed by taking the results estimated for the λi 283

parameter, and considering the dispersion of the results and total penalties according to the modified DPC. 284

The obtained results are presented in Figure 3. A “box and whiskers” plot of the λ ratio for the fR1 and fR3 is 285

represented in Figure 3b. The box plot diagram graphically depicts the statistical five-number summary, 286

consisting of the minimum and maximum values, and the lower (Q1), median (Q2) and upper (Q3) quartiles. 287

Table 3 resumes the obtained results. As expected, a relatively high dispersion was obtained for the 288

predictions of both parameters, which is intrinsically dependent of the dispersion of results in the DB for the 289

fR1 and fR3, since the values of these parameters are also affected by the properties of the surrounding cement 290

matrix, but not considered in the present approach due to the reasons already pointed out. The authors are 291

doing an effort for increasing the database on the characterization of the post-cracking behaviour of SFRC, in 292

order to derive more reliable equations for the determination of fRi. 293

In a design context of a SFRC slab, three point notched SFRC beam bending tests should be executed 294

according to the recommendations of CEB-FIP Model Code 2010 in order to obtain the fRi of the SFRC, and 295

these values are directly used in the proposed model for the evaluation of the punching failure load of a 296

SFRC slab supported on columns. 297

298

THE CONTRIBUTION OF FIBRE REINFORCEMENT FOR THE PUNCHING RESISTANCE OF 299

SFRC FLAT SLABS 300

The contribution of fibre reinforcement mechanisms for the punching resistance of SFRC flat slabs, VR,f, has 301

been investigated by several researchers (Narayanan and Darwish 1987; Harajli et al. 1995; Muttoni and 302

13

Ruiz 2010; Choi et al. 2007; Higashiyama et al. 2011), but none of them has acquired generalized 303

conclusions to be considered as design guideline criteria. 304

With the aim of contributing for the development of a formulation that is sufficiently simple to be adopted in 305

the design practice, and with scientific rigour capable of simulating with enough accuracy a phenomena that 306

has a brittle character and huge impact if a collapse occurs (Gardner et al. 2002), a new approach to 307

determine VR,f is described in this section by combining the most comprehensible knowledge available. This 308

formulation is based on the principles proposed by Muttoni and Ruiz (2010), being the contribution of fibre 309

reinforcement mechanisms simulated by a stress vs crack width law, σf(w), recommended by the CEB-FIP 310

Model Code 2010, but considering Eqs. (20) and (21) to determine σf(w). 311

According to Muttoni and Ruiz (2010), it is acceptable to consider that a slab with axisymmetric structural 312

conditions, when submitted to a load level corresponding to the failure state, can be regarded as a group of 313

radial segments that rotate as rigid bodies, Figure 4. 314

The reinforcement mechanisms offered by the fibres crossing the critical punching surface are simulated by 315

the stress-crack width relationship (Figure 5a), and after convenient integration of σf(w) at the fracture 316

surface, the fiber reinforcement contribution for the punching resistence VR,f can be obtained. 317

According to the Critical Shear Crack Theory (CSCT) (Muttoni 2008; Muttoni and Schwartz 1991), the 318

crack opening of the punching failure crack, w, is proportional to the rotation of the slab, ψ, and the distance 319

from the bottom surface of the slab, z (Figure 5b): 320

( ) zz,w ⋅⋅= ψµψ (23) 321

where µ is the coefficient relating the rotation ψ with the crack opening w. The µ parameter was obtained by 322

using the rotation values at punching failure load, ψu, of the slabs composing the database introduced in 323

Section “Database (DB)”. For each slab its ψu was evaluated from available experimental data, and assuming 324

that the ultimate crack width, wu, should be in the interval 1.5 to 3.0 mm, for each wu in this interval the 325

corresponding value of the µ parameter is determined from Eq. (23). The wu was determined at the level of 326

the tensile flexural reinforcement, considering z=d-x (Figure 5), where the position of the neutral axis, x, is 327

determined by applying the approach recommended by CEB-FIP Model Code 2010 (Figure 6), where fFtu is 328

obtained from Eq. (16) with wu=2.5 mm, as suggested by this standard. Therefore, for the µ parameter that 329

respect Eq. (23) for the considered wu, it is obtained the contribution of fibre reinforcement for the punching 330

14

resistance, VR,f, and applying Eq. (14) the punching failure load is estimated and compared to the value 331

registered experimentally. This algorithm was executed for the adopted interval of wu, and the pair of wu and 332

µ parameters that have best predicted the punching failure load of the experimental programs collected in the 333

database was µ=2.5 and wu=2.5mm (Moraes Neto 2013). 334

According to Moraes Neto (2013) ψu is calculated from the following equation: 335

stsR

u rIE

m. ⋅

−

⋅⋅= χψ

1350 (24)

336

where, 337

⋅−⋅

−⋅⋅⋅⋅=⋅d

x

d

xdEIE s 3

1131 βρ (25)

338

and 339

150

6

1

IE

m.

hE

f cr

s

ctts ⋅

⋅≅⋅

⋅⋅⋅

=βρ

χ (26)

340

In equation (25), E·I1 represents the flexural stiffness of SFRC cracked cross section, obtained according to 341

the procedures adopted for RC members (Moraes Neto 2013), and assuming a stabilized cracking stage. The 342

contribution of fibre reinforcement for the E·I1 is only indirectly taken in the evaluation of the neutral axis, x, 343

Figure 6. In Eq. (25) β is a factor to take into account the real arrangement of the reinforcement, since the 344

CSCT is supported on the principle of axisymmetric structural conditions, but the majority of the built and 345

tested RC flat slabs have orthogonal distribution of the reinforcement (Guandalini 2005). According to 346

Muttoni (2008), β=0.6 yields to satisfactory results. The evaluation of the position of the neutral axis, x, was 347

made according to the recommendations of CEB-FIP Model Code 2010, see Figure 6. 348

The χts factor in Eq. (26) simulates the post-cracking tensile strength of cracked concrete (tension stiffening 349

effect), where fct is the concrete tensile strength, Es is the elasticity modulus of the steel reinforcement, ρ is 350

the reinforcement ratio of the tensile flexural reinforcement, h is the slab thickness, and mcr= fct h2/6 is the 351

cracking moment. 352

To evaluate VR,f it is assumed that the post-cracking stress law, σf(w)=σf(ψ,z), can be represented by the 353

following linear constitutive σ-w approach recommended by CEB-FIP Model Code 2010: 354

15

( ) ( ) ( ) 0205052 13 ≥⋅+⋅−⋅−== RRFtsFtsFtuf f.f.f.

wfwfwσ (27)

355

Replacing Eq. (23) into Eq. (27) yields: 356

( ) ( ) 0205052 13 ≥⋅+⋅−⋅⋅⋅−= RRFtsFtsf f.f.f.

zfz,

ψµψσ (28)

357

The VR,f is obtained by integrating σf(ψ,z) on the area A0, where A0, see Figure 5a, represents the horizontal 358

projection of the punching failure surface (Moraes Neto 2013): 359

( )∫ ⋅=0

0A

ff,R dAz,V ψσ (29)

( ) ( ) ( ) ( ) ( )( )

⋅−⋅−+⋅−⋅−⋅Ω⋅Ω⋅⋅

−⋅Ω⋅Ω−Ω⋅Ω⋅+Ω⋅Ω⋅⋅⋅−⋅⋅=

k

kkkkd

dkdV f,R

213

211422

3241

31221 32

ψ

ψπψ (30)

360

where, 361

Ftsf=Ω1 ; ( )

52

20502 13

.

f.f.f RRFts ⋅+⋅−⋅=Ω µ; cr=Ω3 ; ( )k−⋅

=Ω12

14 (31)

stsR

u rIE

m. ⋅

−

⋅⋅== χψψ

1350 ;

d

xk = (32)

362

To evaluate the VR,f an assumption was assumed by considering σf(ψ,z)=σf(w)=σf(wu) (see Eq. (23)) and 363

adopting for wu the value 1.5 mm recommended by CEB-FIP Model Code 2010 (clause 7.7.3.5.3). 364

Therefore: 365

( ) ( ) ( ) 000

00

AwdAwdAz,V ufA

ufA

ff,R ⋅=⋅=⋅= ∫∫ σσψσ (33)

366

where σf(wu) is obtained from Eq. (16): 367

( ) ( ) ( ) 0205052 13 ≥⋅+⋅−⋅−== RRFtsu

FtsuFtuuf f.f.f.

wfwfwσ (34)

368

resulting: 369

( ) 013 205052

51Af.f.f

.

.fV RRFtsFtsf,R ⋅

⋅+⋅−⋅−= (35)

370

16

ASSESSMENT OF THE PREDICTIVE PERFORMANCE OF FORMULATIONS FOR SFRC FLAT 371

SLABS 372

Since the formulation of CEB-FIP Model Code 2010 has best fitted the VR,c of the collected DB, it was 373

selected to be coupled with the proposed model that predicts the contribution of fibre reinforcement for the 374

punching resistance of SFRC flat slabs, VR,f, resulting a model capable of estimating the the punching failure 375

load of SFRC slabs reinforced with longitudinal steel bars: Vthe=VR=VR,c+VR,f. In the previous Section two 376

equations were proposed to estimate VR,f: i) Eq. (30) that requires performing a full integration of the crack 377

opening–fibre stress law, thereby is herein designated as The-refin; 2) Eq. (35) that is more simple to obtain, 378

thereby is herein designated as The-simpl, where refin and simpl means refined and simplified, respectively. 379

In Table 4 the predictive performance of these two models is compared to the one resulting from the 380

application of the formulation proposed by CEB-FIP Model Code 2010 that was already described. It can be 381

concluded that both proposed formulations evidence excellent predictive performance with a relatively small 382

COV. Both formulations present average value of λ much closer to the unit value than the CEB-FIP model, 383

and lower COV. Furthermore, these models have also the largest number of predictions in the appropriate 384

safety interval according to the DPC (Table 1). Based on the results and considering its easy attainement, it is 385

recommended to use The-simpl approach to predict VR,f, based on Eq. (35). 386

The CEB-FIP Model Code 2010 provides a large number of predictions against safety, e.g. a λ value in the 387

interval [0.50-0.85[, considered “Dangerous” according to the DPC (see Table 1), was obtained for 39 slabs. 388

In the CEB-FIP Model Code 2010 a constant post-cracking residual strength is assumed distributed in the 389

punching fracture surface for the simulation of the fibre reinforcement contribution for the punching 390

resistance. This is in fact the same strategy adopted in the simplified approach herein proposed (Eq. (35)), 391

but the relatively high number of unsafe predictions demonstrates that the punching failure surface assumed 392

in this standard (b0.d) seems not realistic, or not compatible with the assumption of a constant residual 393

strength distribution in the punching failure surface. In this model the VR,f is calculated from fFtu determined 394

by Eq. (16). To evaluate fFtu, instead of adopting wu=ψ·d/6 as recommended by this standard, it was assumed 395

wu=1.5 mm, since the former approach lead to more unsafe predictions. If a proper safety criterion is 396

considered as the one that 85% of the samples remains in the interval λ= [0.85-1.15[, a wu≈4.0 mm should be 397

adopted in the model proposed by CEB-FIP Model Code 2010. 398

17

399

COMPARISON OF THE PREDICTIVE PERFORMANCE OF THE DEVELOPED MODELS 400

In this section the predictive performance of the refined and simplified models (The-refin, The-simpl) 401

developed in the present work for the evaluation of the punching failure load of SFRC flat slabs is compared 402

to the one of the following models found in the literature: Narayanan and Darwish (1987), Shaaban and 403

Gesund (1994), Harajli et al. (1995), Holanda (2002), Choi et al. (2007), Muttoni and Ruiz (2010) and 404

Higashiyama et al. (2011). The formulation of these models is presented in Moraes Neto (2013) and Moraes-405

Neto et al. (2012). Like in a previous Section, the predictive performance of the models was based on the 406

evaluation of the λ=Vexp/Vthe parameter and on the analysis of λ according to the modified version of the 407

DPC, where the Vexp is the punching failure load of the slabs collected in the database described in Section 408

“Database (DB)”. The models in analysis are designated as MODi (i=1 up to 9), with the corresponding 409

formulation assigned in the footnote of Table 5 and in the caption of Figure 7. The box plot diagram in 410

Figure 7 graphically depicts the statistical five-number summary, consisting of the minimum and maximum 411

values, and the lower, median and upper quartiles of λ for each model. From the analysis of Figure 7 and the 412

values included in Table 5 it can be concluded that the proposed models, together with the model of Muttoni 413

and Ruiz (2010), are those that assure values of λ closer to the unity with the lowest COV. However, the 414

models proposed in the present work provided the smallest total penalties, with the largest number of 415

predictions in the appropriate safety interval. 416

417

CONCLUSIONS 418

In the present paper a model was proposed to predict the punching failure load of steel fibre reinforced 419

concrete (SFRC) slabs centrically loaded (VR). This model is supported on the assumption that VR=VR,c+VR,f, 420

where VR,c and VR,f is the contribution of concrete and fibre reinforcement for the punching resistance, 421

respectively. To determine the best available formulation for the prediction of VR,c, the predictive 422

performance of models proposed by ACI 318, CEB-FIP Model Code 1990, EC2 and CEB-FIP Model Code 423

2010 was assessed by estimating the punching tests results collected in a database (DB) built for this 424

purpose. From this study it was concluded that CEB-FIP Model Code 2010 evaluates more accurately the 425

concrete contribution for the punching resistance of RC slabs, and consequently it was selected to be 426

18

combined with the formulation developed for the prediction of the punching resistance of SFRC slabs. This 427

formulation is supported in the critical shear crack theory, and integrates a stress-crack width relationship 428

(σf(w)) for modelling the contribution of fibre reinforcement mechanisms. The σf(w) was determined 429

according to the recommendations of CEB-FIP Model Code 2010 for the characterisation of the post-430

cracking flexural tensile behaviour of FRC. The proposed model has two levels of sophistication, one of 431

more laborious calculus, and the other with a simpler way of obtaining the VR,f. 432

The predictive performance of these two versions of the developed model was appraised by simulating the 433

punching tests composing the DB. Both versions of the model have predicted with high accuracy the failure 434

load of the punching tests of the DB, and assured better and safer predictions than the ones obtained with 435

available models for the evaluation of the punching failure load of SFRC slabs. 436

437

ACKNOWLEDGEMENTS 438

The study presented in this paper is a part of the research project titled “SlabSys-HFRC - Flat slabs for multi-439

storey buildings using hybrid reinforced self-compacting concrete: an innovative structural system”, with 440

reference number of PTDC/ECM/120394/2010. The first author acknowledges the support provided by the 441

CAPES and CNPq grant, and the grant provided by the project SlabSys. 442

443

REFERENCES 444

ACI 318. Building Code Requirements for Structural Concrete, American Concrete Institute, Farmington 445

Hills, Michigan, 2008. 446

Barros, J.A.O.; Cunha, V.M.C.F.; Ribeiro, A.F.; Antunes, J.A.B. “Post-Cracking Behaviour of Steel Fibre 447

Reinforced Concrete”. RILEM Materials and Structures Journal, 38(275), pp. 47-56, 2005. 448

Barros, J.A.O.; Salehian, H.; Pires, N.M.M.A; Gonçalves, D.M.F. “Design and testing elevated steel fibre 449

reinforced self-compacting concrete slabs”. BEFIB2012–Fibre reinforced concrete, 2012. 450

BS 8110. Structural Use of Concrete. British Standards Institution, 1985. 451

CEB-FIP. Model Code 1990: Final Draft. Bulletin D’Information, n° 203-205, CEB, Lausanne, July 1991. 452

CEB-FIP. Model Code 2010: Final Draft. Model Code prepared by Special Activity Group 5, Lausanne, 453

September 2011. 454

19

Choi, K-K.; Taha, M.M.R.; Park, H-G.; Maji, A.K. “Punching shear strength of interior concrete slab-455

column connections reinforced with steel fibers”. Cement & Concrete Composites, 29(5), pp. 409-420, May, 456

2007. 457

Collins, M.P. “Evaluation of shear design procedures for concrete structures”. A Report prepared for the 458

CSA technical committee on reinforced concrete design, 2001. 459

Cunha, V.M.C.F.; Barros, J.A.O.; Sena-Cruz, J.M. “Pullout behaviour of steel fibres in self-compacting 460

concrete”. ASCE Materials in Civil Engineering Journal, 22(1), pp. 1-9, January, 2010. 461

Destrée, X.; Mandl, J. “Steel fibre only reinforced concrete in free suspended elevated slabs: Case studies, 462

design assisted by testing route, comparison to the latest SFRC standard documents”. Tailor Made Concrete 463

Structures – Walraven & Stoelhorst (eds), Taylor & Francis Group, pp. 437-443, London, 2008. 464

Destrée, X.; Bissen, A.M.; Bissen, Luxembourg. “Steel-fibre-only reinforced concrete in free suspended 465

elevated slabs”. Concrete Engineering International, Spring, pp. 47-49, 2009. 466

Espion, B. “Test report n°33396”, University of Brussels, Belgium, 2004. 467

Eurocode 2. Design of Concrete Structures. Part 1-1: General Rules and Rules for Buildings. European 468

Standard, 2004. 469

Gardner, N.J.; Huh, J.; Chung, L. “Lessons from the Sampoong department store collapse”. Cement & 470

Concrete Composites, 24, pp. 523-529, 2002. 471

Guandalini, S. Poinçonnement symétrique des dalles en béton armé. PhD Thesis 3380, Ecole Polytechnique 472

Fédérale de Lausanne, Switzerland, 2005. (in French) 473

Harajli, M.H.; Maalouf D.; Khatib H. “Effect of fibrs on the punching shear strength of slab-column 474

connections”. Cement & Concrete Composites, 17(2), pp. 161-170, 1995. 475

Higashiyama, H.; OTA, A.; Mizukoshi, M. “Design equation for punching shear capacity of SFRC slabs”. 476

International Journal of Concrete Structures and Materials, 5(1), pp. 35-42, 2011. 477

Holanda, K.M.A. Analysis of resistant mechanisms and similarities of the adition effect of steel fibers on 478

strength and ductility to both punching shear of flat and the shear of concrete beams. PhD Thesis. São 479

Carlos, Brazil, 2002. (in Portuguese) 480

Johansen, K.W. “Yield-line theory”. Cement and Concrete Association, 1962. 481

Mandl, J. “Flat slabs made of steel fibre reinforced concrete (SFRC),” CPI Worldwide, 1, 2008. 482

20

Michels, J.; Waldmann, D.; Maas, S.; Zürbes, A. “Steel fibers as only reinforcement for flat slab construction 483

– Experimental investigation and design”. Construction and Building Materials, 26(1), pp. 145-155, 2012. 484

Moraes-Neto, B.N.; Barros, J.A.O.; Melo, G.S.S.A. “The predictive performance of design models for the 485

punching resistance of SFRC slabs in inner column loading conditions”. 8th RILEM International symposium 486

on fibre reinforced concrete: challenges and opportunities (BEFIB 2012), September, 2012. 487

Moraes Neto, B.N. Punching behaviour of steel fibre reinforced concrete slabs submitted to symmetric 488

loading. PhD in Civil Engineering, Department of Civil and Environmental Engineering, University of 489

Brasília, Brasília, DF, January, 2013. (in Portuguese) 490

Muttoni, A.; Schwartz, J. “Behaviour of Beams and Punching in Slabs without Shear Reinforcement”. IABSE 491

Colloquium, 62, pp. 703-708, Zurich, Switzerland, 1991. 492

Muttoni, A. “Punching shear strength of reinforced concrete slabs without transverse reinforcement”. ACI 493

Structural Journal, 105(4), pp. 440-450, July/August, 2008. 494

Muttoni, A.; Ruiz, M.F. “The critical shear crack theory as mechanical model for punching shear design and 495

its application to code provisions”. Fédération Internationale du Béton, Bulletin 57, Lausanne, Switzerland, 496

pp. 31-60, 2010. 497

Narayanan, R., Darwish, I.Y.S. “Punching Shear Tests on Steel Fibre Reinforced Microconcrete Slabs”, 498

Magazine of Concrete Research, 39(138), pp. 42-50, 1987. 499

Sasani, M. and Sagiroglu, S. “Progressive collapse of reinforced concrete structures: a multihazard 500

perspective”, ACI Structural J., 105(1), pp. 96-105, 2008. 501

Shaaban, A.M.; Gesund, H. “Punching shear strength of steel fiber reinforced concrete flat plates”. ACI 502

Structural Journal, 91(4), pp. 406-414, Jul/Aug, 1994. 503

504

505

506

507

508

509

510

21

511

512

513

NOTATION 514

A0 Horizontal projection of failure surface

As Area of tension reinforcement

A’s Area of compression reinforcement

b Width of a isolated slab element

b0 Critical perimeter for punching shear (ACI 318 and CEB-FIP Model Code 2010)

bs Strip of slab to avaluet the bending moment

d Internal arm of the slab

df Diameter of fibre

dg Maximum diameter of the

E Modulus of elasticity of concrete

Es Modulus of elasticity of reinforcement

fc Average compressive strength of concrete in cylinder specimens

fct Average tensile strength of concrete (Brazilian test)

fFts Post-cracking strength for serviceability crack opening

fFtu Post-cracking strength for ultimate crack opening

fRi Residual flexural tensile strength of fibre reinforced concrete corresponding to CMODi

fyd Design yield strength of reinforcement

Fs Internal compressive force of tensile reinforcement

F’ s Internal compressive force of compressive reinforcement

h Slab thickness

I1 Second moment of area of cracked concrete cross-section

k, ξ Size effect parameter

kdg Aggregate interlock parameter

kψ Rotation of the slab parameter

l f Length of fibre

L Span of slab

mR Resisting bending moment (plastic bending moment)

msd Actuating bending moment

r0 Radius of the critical shear crack

rc Radius of a circular column

rq Radius of the load introduction at the perimeter

rs Radius of circular isolated slab element

u1 Critical perimeter for punching shear (EC2 and CEB-FIP Model Code 1990)

V Shear force

Vf Fibre volume percentage

VR,cd Design concrete contribution to punching shear strength

VR,d Design punching shear strength

VR,fd Design fibre contribution to punching shear strength

VR,sd Design shear reinforcement contribution to punching shear strength

Vs,d Actuating shear force

w Shear crack opening

wu Maximum acceptable crack width imposed by design conditions

x Neutral axis of slab

22

αs Parameter for columns located in the interior of the building

β Efficiency factor of the bending reinforcement for stiffness calculation

βc Ratio between the larger and the smaller edge of the column’s cross section

χts Tension stiffening parameter

εc Concrete strain

εcu Ultimate strain of concrete in compression zone

εfu Ultimate strain of fibre in tensile zone

εs Strain of steel reinforcement in tensile zone

εsu Ultimate strain of steel reinforcement in tensile zone

ε’ s Compressive steel reinforcement strain

εt,bot Concrete tensile strain at the bottom surface of the slab

νN,d Design nominal shear stress

νR,d Design shear strength

τf Average interracial bond strength of fibre matrix

ρ Tensile reinforcement ratio

σ Stress

ψ Rotation of slab

515

23

516 LIST OF TABLE CAPTIONS 517

518

Table 1. Modified version of the Demerit Points Classification (DPC). 519

Table 2. Predictive performance of the design models according to the modified version of the DPC. 520

Table 3. Predictive performance of Eqs. (20) and (21) in the context of the modified version of the DPC. 521

Table 4. Performance of models for the prediction of the punching failure load of SFRC flat slabs according to the 522

modified version of the DPC. 523

Table 5. Performance of several models to predict Vexp: classification of the models according to the modified version of 524

the DPC 525

526

Table 1. Modified version of the Demerit Points Classification (DPC).

λ=Vexp/Vthe Classification Penalty (PEN)

< 0.50 Extremely Dangerous 10

[0.50-0.85[ Dangerous 5

[0.85-1.15[ Appropriate Safety 0

[1.15-2.0[ Conservative 1

≥ 2.0 Extremely Conservative 2

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

24

542

Table 2. Predictive performance of the design models according to the modified version of the DPC.

λ=Vexp/Vthe < 0.50 [0.50-0.85[ [0.85-1.15[ [1.15-2.00[ ≥ 2.00 Total PEN

AVG STD COV (%)

MOD1 N° samples 0 3 3 18 0 24

1.28 0.32 25.28 PEN 0 15 0 18 0 33

MOD2 N° samples 0 7 14 3 0 24

0.95 0.20 20.91 PEN 0 35 0 3 0 38

MOD3 N° samples 0 1 15 8 0 24

1.16 0.25 21.30 PEN 0 5 0 8 0 13

MOD4 N° samples 0 2 21 1 0 24

1.01 0.09 9.34 PEN 0 10 0 1 0 11

MOD1=ACI 318, MOD2 =CEB-FIP Model Code 1990, MOD3=EC2, MOD4=CEB-FIP Model Code 2010.

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

25

564

Table 3. Predictive performance of Eqs. (20) and (21) in the context of the modified version of the DPC.

fRi fR1 fR3

λi=fRi,exp/fRi,the N° samples PEN N° samples PEN

< 0.50 0 0 1 10

[0.50-0.85[ 4 20 7 35

[0.85-1.15[ 17 0 18 0

[1.15-2.00[ 43 43 38 38

≥ 2.00 5 10 5 10

Total PEN 69 73 69 93

Statistical resume

fRi fR1 fR3

Average (AVG) 1.37 1.32

STD 0.38 0.48

COV (%) 27.88 36.08

565

566

567

568

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

26

584

Table 4. Performance of models for the prediction of the punching failure load of SFRC flat slabs according to the modified version of the DPC.

Models The-refin The-simpl CEB-FIP Model Code 2010

λ=Vexp /Vthe N° samples PEN N° samples PEN N° samples PEN

< 0.50 0 0 0 0 2 20

[0.50-0.85[ 6 30 5 25 39 195

[0.85-1.15[ 43 0 42 0 9 0

[1.15-2.00[ 1 1 3 3 0 0

≥ 2.00 0 0 0 0 0 0

Total PEN 50 31 50 28 50 215

Statistical resume

Models The-refin The-simpl CEB-FIP Model Code2010 Average (AVG)

0.97 0.98 0.73

STD 0.11 0.11 0.13

COV (%) 11.38 11.17 17.48

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

27

603

Table 5. Performance of several models to predict Vexp: classification of the models according to the modified version of the DPC

λ=Vexp /Vthe < 0.50 [0.50-0.85[ [0.85-1.15[ [1.15-2.00[ ≥ 2.00 Total PEN AVG STD COV (%)

MOD1 N° samples 0 21 21 8 0 50

0.92 0.23 25.29 PEN 0 105 0 8 0 113

MOD2 N° samples 0 2 18 29 1 50

1.24 0.26 20.89 PEN 0 10 0 29 2 41

MOD3 N° samples 0 5 18 20 7 50

1.42 0.62 43.38 PEN 0 25 0 20 14 59

MOD4 N° samples 0 0 8 42 0 50

1.32 0.20 15.47 PEN 0 0 0 42 0 42

MOD5 N° samples 0 6 17 27 0 50

1.20 0.29 24.03 PEN 0 30 0 27 0 57

MOD6 N° samples 0 6 37 7 0 50

0.99 0.13 13.26 PEN 0 30 0 7 0 37

MOD7 N° samples 0 20 24 6 0 50

0.92 0.18 19.45 PEN 0 100 0 6 0 106

MOD8 N° samples 0 6 43 1 0 50

0.97 0.11 11.38 PEN 0 30 0 1 0 31

MOD9 N° samples 0 5 42 3 0 50

0.98 0.11 11.17 PEN 0 25 0 3 0 28

MOD1= Narayanan and Darwish (1987); MOD2= Shaaban and Gesund (1994); MOD3= Harajli et al. (1995); MOD4= Holanda (2002); MOD5= Choi et al. (2007); MOD6= Muttoni and Ruiz (2010); MOD7= Higashiyama et al. (2011); MOD8=The-refin; MOD9=The-simpl

604

605

606

607

608

609

28

610

611

612

29

613

614

30

615

616

617

618

619

620

621

622